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Chapter 7

Geodesics on Riemannian

7.1 , Local Existence and Uniqueness

If (M,g)isaRiemannianmanifold,thentheconceptof makes sense for any piecewise smooth (in fact, C1) on M.

Then, it possible to define the structure of a on M,whered(p, q)isthegreatestlowerboundofthe length of all joining p and q.

Curves on M which locally yield the shortest between two points are of great interest. These curves called geodesics play an important role and the goal of this chapter is to study some of their properties.

489 490 CHAPTER 7. GEODESICS ON RIEMANNIAN MANIFOLDS

Given any p M,foreveryv TpM,the(Riemannian) norm of v,denoted∈ v ,isdefinedby∈ " " v = g (v,v). " " p ! The Riemannian inner product, gp(u, v), of two vectors, u, v TpM,willalsobedenotedby u, v p,or simply u, v .∈ # $ # $

Definition 7.1.1 Given any Riemannian , M, a smooth parametric curve (for short, curve)onM is amap,γ: I M,whereI is some open of R. For a closed→ interval, [a, b] R,amapγ:[a, b] M is a smooth curve from p =⊆γ(a) to q = γ(b) iff→γ can be extended to a smooth curve γ:(a ", b + ") M, for some ">0. Given any two points,− p, q →M,a ∈ continuous , γ:[a, b] M,isa" piecewise smooth curve from p to q iff →

(1) There is a sequence a = t0

The set Ω(M; p, q)isanimportantobjectsometimes called the path space of M (from p to q).

Unfortunately it is an infinite-dimensional manifold, which makes it hard to investigate its properties.

Observe that at any junction , γi 1(ti)=γi(ti), there may be a jump in the velocity vector− of γ.

We let γ(((ti)+)=γi((ti)andγ(((ti) )=γi( 1(ti). − − 492 CHAPTER 7. GEODESICS ON RIEMANNIAN MANIFOLDS Given any curve, γ Ω(M; p, q), the length, L(γ), of γ is defined by ∈ k 1 − ti+1 L(γ)= γ((t) dt " " i=0 $ti #k 1 − ti+1 = g(γ((t),γ((t)) dt. i=0 ti # $ % It is easy to see that L(γ)isunchangedbyamonotone reparametrization (that is, a map h:[a, b] [c, d], whose → , h(,hasaconstantsign).

Let us now assume that our , (M,g), is equipped with the Levi-Civita and thus, for D every curve, γ,onM,letdt be the associated along γ,alsodenoted ∇γ( 7.1. GEODESICS, LOCAL EXISTENCE AND UNIQUENESS 493 Definition 7.1.2 Let (M,g)beaRiemannianmani- fold. A curve, γ: I M,(whereI R is any interval) → ⊆ is a iff γ((t)isparallelalongγ,thatis,iff

Dγ( = γ( =0. dt ∇γ(

If M was embedded in Rd,ageodesicwouldbeacurve, Dγ( γ,suchthattheaccelerationvector,γ(( = dt ,isnormal to Tγ(t)M.

By Proposition 6.4.6, γ((t) = g(γ (t),γ(t)) is con- " " ( ( stant, say γ((t) = c. " " % 494 CHAPTER 7. GEODESICS ON RIEMANNIAN MANIFOLDS If we define the arc-length , s(t), relative to a, where a is any chosen point in I,by t s(t)= g(γ (t),γ(t)) dt = c(t a),tI, ( ( − ∈ $a we conclude that% for a geodesic, γ(t), the parameter, t,is an affine function of the arc-length.

The geodesics in Rn are the straight lines parametrized by constant velocity.

The geodesics of the 2- are the great , parametrized by arc-length.

The geodesics of the Poincar´ehalf- are the lines x = a and the half-circles centered on the x-axis.

The geodesics of an ellipsoid are quite fascinating. They can be completely characterized and they are parametrized by elliptic functions (see Hilbert and Cohn-Vossen [?], Chapter 4, Section and Berger and Gostiaux [?], Section 10.4.9.5). 7.1. GEODESICS, LOCAL EXISTENCE AND UNIQUENESS 495 If M is a of Rn,geodesicsarecurveswhose acceleration vector, γ(( =(Dγ()/dt is to M (that is, for every p M, γ(( is normal to T M). ∈ p In a local chart, (U,ϕ), since a geodesic is characterized by the fact that its velocity vector field, γ((t), along γ is , by Proposition 6.3.4, it is the solution of the following system of second-order ODE’s in the unknowns, uk: d2u du du k + Γk i j =0,k=1,...,n, dt2 ij dt dt ij # with u = pr ϕ γ (n =dim(M)). i i ◦ ◦ 496 CHAPTER 7. GEODESICS ON RIEMANNIAN MANIFOLDS The standard existence and uniqueness results for ODE’s can be used to prove the following proposition:

Proposition 7.1.3 Let (M,g) be a Riemannian man- ifold. For every point, p M,andeverytangentvec- ∈ tor, v TpM,thereissomeinterval,( η,η),anda unique∈ geodesic, − γ :( η,η) M, v − → satisfying the conditions

γv(0) = p, γv( (0) = v.

The following proposition is used to prove that every geodesic is contained in a unique maximal geodesic (i.e, with largest possible domain): 7.1. GEODESICS, LOCAL EXISTENCE AND UNIQUENESS 497 Proposition 7.1.4 For any two geodesics, γ : I M and γ : I M,ifγ (a)=γ (a) and 1 1 → 2 2 → 1 2 γ1( (a)=γ2( (a),forsomea I1 I2,thenγ1 = γ2 on I I . ∈ ∩ 1 ∩ 2

Propositions 7.1.3 and 7.1.4 imply that for every p M and every v T M,thereisauniquegeodesic,denoted∈ ∈ p γv,suchthatγ(0) = p, γ((0) = v,andthedomainofγ is the largest possible, that is, cannot be extended.

We call γv a maximal geodesic (with initial conditions γv(0) = p and γv( (0) = v).

Observe that the system of differential satisfied by geodesics has the following homogeneity property: If t γ(t)isasolutionoftheabovesystem,thenforevery constant,,→ c,thecurvet γ(ct)isalsoasolutionofthe system. ,→ 498 CHAPTER 7. GEODESICS ON RIEMANNIAN MANIFOLDS We can use this fact together with standard existence and uniqueness results for ODE’s to prove the proposition below.

Proposition 7.1.5 Let (M,g) be a Riemannian man- ifold. For every point, p0 M,thereisanopensub- set, U M,withp U∈,andsome">0,sothat: ⊆ 0 ∈ For every p U and every , v TpM, with v <"∈,thereisauniquegeodesic, ∈ " " γ :( 2, 2) M, v − → satisfying the conditions

γv(0) = p, γv( (0) = v.

If γ :( η,η) M is a geodesic with initial conditions v − → γv(0) = p and γv( (0) = v =0,foranyconstant,c =0,the curve, t γ (ct), is a geodesic- defined on ( η-/c, η/c) ,→ v − (or (η/c, η/c)ifc<0) such that γ((0) = cv.Thus, − γ (ct)=γ (t),ct( η,η). v cv ∈ − This fact will be used in the next section. 7.2. THE EXPONENTIAL MAP 499 7.2 The Exponential Map

The idea behind the exponential map is to parametrize aRiemannianmanifold,M,locallynearanyp M ∈ in terms of a map from the TpM to the manifold, this map being defined in terms of geodesics.

Definition 7.2.1 Let (M,g)beaRiemannianmani- fold. For every p M,let (p)(orsimply, )bethe ∈ D D open subset of TpM given by (p)= v T M γ (1) is defined , D { ∈ p | v } where γv is the unique maximal geodesic with initial con- ditions γv(0) = p and γv( (0) = v.Theexponential map is the map, exp : (p) M,givenby p D → expp(v)=γv(1).

It is easy to see that (p)isstar-shaped,whichmeans that if w (p), thenD the segment tw 0 t 1 is contained∈D in (p). { | ≤ ≤ } D 500 CHAPTER 7. GEODESICS ON RIEMANNIAN MANIFOLDS In view of the remark made at the of the previous section, the curve t exp (tv),tv(p) ,→ p ∈D is the geodesic, γv,throughp such that γv( (0) = v.Such geodesics are called radial geodesics .Thepoint,expp(tv), is obtained by running along the geodesic, γv,anarc length equal to t v ,startingfromp. " " In general, (p)isapropersubsetofT M. D p Definition 7.2.2 ARiemannianmanifold,(M,g), is geodesically complete iff (p)=T M,forallp M, D p ∈ that is, iffthe exponential, expp(v), is defined for all p M and for all v T M. ∈ ∈ p

Equivalently, (M,g)isgeodesicallycompleteiffevery geodesic can be extended indefinitely.

Geodesically complete manifolds have nice properties, some of which will be investigated later.

Observe that d(expp)0 =idTpM . 7.2. THE EXPONENTIAL MAP 501

It follows from the theorem that expp is adiffeomorphismfromsomeopenballinTpM centered at 0 to M.

The following slightly stronger proposition can be shown:

Proposition 7.2.3 Let (M,g) be a Riemannian man- ifold. For every point, p M,thereisanopensubset, W M,withp W and∈ a number ">0,sothat ⊆ ∈ (1) Any two points q1,q2 of W are joined by a unique geodesic of length <".

(2) This geodesic depends smoothly upon q1 and q2,

that is, if t expq1(tv) is the geodesic joining q and q (0 ,→ t 1), then v T M depends 1 2 ≤ ≤ ∈ q1 smoothly on (q1,q2). (3) For every q W ,themapexp is a diffeomor- ∈ q phism from the open , B(0,") TqM,toits , U =exp(B(0,")) M,with⊆ W U and q q ⊆ ⊆ q Uq open. 502 CHAPTER 7. GEODESICS ON RIEMANNIAN MANIFOLDS For any q M,anopenneighborhoodofq of the form, ∈ Uq =expq(B(0,")), where expq is a diffeomorphism from the open ball B(0,")ontoUq,iscalledanormal neigh- borhood .

Definition 7.2.4 Let (M,g)beaRiemannianmani- fold. For every point, p M,theinjectivity radius of M at p,denotedi(p), is∈ the least upper bound of the numbers, r>0, such that expp is a diffeomorphism on the open ball B(0,r) TpM.Theinjectivity radius, i(M),ofM is the greatest⊆ lower bound of the numbers, i(p), where p M. ∈

For every p M,wegetachart,(Up,ϕ), where ∈ 1 Up =expp(B(0,i(p))) and ϕ =exp− ,calledanormal chart.

If we pick any orthonormal , (e1,...,en), of TpM, 1 then the x ’s, with x = pr exp− and pr the i i i ◦ i onto Rei,arecallednormal coordinates at p (here, n =dim(M)).

These are defined up to an of TpM. 7.2. THE EXPONENTIAL MAP 503 The following proposition shows that Riemannian metrics do not admit any local invariants of order one:

Proposition 7.2.5 Let (M,g) be a Riemannian man- ifold. For every point, p M,innormalcoordinates at p, ∈ ∂ ∂ g , = δ and Γk (p)=0. ∂x ∂x ij ij & i j 'p

For the next proposition, known as Gauss Lemma,we need to define polar coordinates on TpM.

If n =dim(M), observe that the map, n 1 (0, ) S − T M 0 ,givenby ∞ × −→ p −{ } n 1 (r, v) rv, r > 0,v S − ,→ ∈ n 1 is a diffeomorphism, where S − is the sphere of radius r =1inTpM. 504 CHAPTER 7. GEODESICS ON RIEMANNIAN MANIFOLDS

n 1 Then, the map, f:(0,i(p)) S − U p ,givenby × → p −{ } n 1 (r, v) exp (rv), 0

Proposition 7.2.6 (Gauss Lemma)Let(M,g) be a Riemannian manifold. For every point, p M,the images, exp (S(0,r)),ofthespheres,S(0,r)∈ T M, p ⊆ p centered at 0 by the exponential map, expp,areorthog- onal to the radial geodesics, r expp(rv),throughp, for all r

ω(t)=expp(r(t)v(t)), with 0

Proposition 7.2.7 Let (M,g) be a Riemannian man- ifold. We have b ω((t) dt r(b) r(a) , " " ≥| − | $a where equality holds only if the function r is monotone and the function v is constant. Thus, the shortest path joining two concentric spherical shells, expp(S(0,r1)) and expp(S(0,r2)),isaradialgeodesic. 506 CHAPTER 7. GEODESICS ON RIEMANNIAN MANIFOLDS We now get the following important result from Proposi- tion 7.2.6 and Proposition 7.2.7:

Theorem 7.2.8 Let (M,g) be a Riemannian mani- fold. Let W and " be as in Proposition 7.2.3 and let γ:[0, 1] M be the geodesic of length <"joining two → points q1,q2 of W .Foranyotherpiecewisesmooth path, ω,joiningq1 and q2,wehave 1 1 γ((t) dt ω((t) dt " " ≤ " " $0 $0 where equality can holds only if the images ω([0, 1]) and γ([0, 1]) coincide. Thus, γ is the shortest path from q1 to q2.

Here is an important consequence of Theorem 7.2.8. 7.2. THE EXPONENTIAL MAP 507 Corollary 7.2.9 Let (M,g) be a Riemannian mani- fold. If ω:[0,b] M is any curve parametrized by arc-length and ω→has length less than or equal to the length of any other curve from ω(0) to ω(b),thenω is ageodesic.

Definition 7.2.10 Let (M,g)beaRiemannianmani- fold. A geodesic, γ:[a, b] M,isminimal iffits length is less than or equal to the→ length of any other piecewise smooth curve joining its endpoints.

Theorem 7.2.8 asserts that any sufficiently small segment of a geodesic is minimal.

On the other hand, a long geodesic may not be minimal. For example, a great arc on the unit sphere is a geodesic. If such an arc has length greater than π,then it is not minimal. 508 CHAPTER 7. GEODESICS ON RIEMANNIAN MANIFOLDS Minimal geodesics are generally not unique. For example, any two antipodal points on a sphere are joined by an infinite number of minimal geodesics.

A broken geodesic is a piecewise smooth curve as in Def- inition 7.1.1, where each curve segment is a geodesic.

Proposition 7.2.11 ARiemannianmanifold,(M,g), is connected iffany two points of M can be joined by abrokengeodesic.

In general, if M is connected, then it is not true that any two points are joined by a geodesic. However, this will be the case if M is geodesically complete, as we will see in the next section.

Next, we will see that a Riemannian metric induces a distance on the manifold whose induced agrees with the original metric. 7.3. COMPLETE RIEMANNIAN MANIFOLDS, HOPF-RINOW, CUT 509 7.3 Complete Riemannian Manifolds, the Hopf-Rinow Theorem and the Cut Locus

Every connected Riemannian manifold, (M,g), is a met- ric space in a natural way.

Furthermore, M is a complete iff M is geodesi- cally complete.

In this section, we explore briefly some properties of com- plete Riemannian manifolds.

Proposition 7.3.1 Let (M,g) be a connected Rieman- nian manifold. For any two points, p, q M,let d(p, q) be the greatest lower bound of the ∈ of all piecewise smooth curves joining p to q.Then,d is ametriconM and the topology of the metric space, (M,d),coincideswiththeoriginaltopologyofM. 510 CHAPTER 7. GEODESICS ON RIEMANNIAN MANIFOLDS The distance, d,isoftencalledtheRiemannian distance on M.Foranyp M and any ">0, the metric ball of ∈ center p and radius " is the subset, B"(p) M,given by ⊆ B (p)= q M d(p, q) <" . " { ∈ | } The next proposition follows easily from Proposition 7.2.3:

Proposition 7.3.2 Let (M,g) be a connected Rieman- nian manifold. For any compact subset, K M, there is a number δ>0 so that any two points,⊆ p, q K,withdistanced(p, q) <δare joined by a unique∈ geodesic of length less than δ.Furthermore, this geodesic is minimal and depends smoothly on its endpoints.

Recall from Definition 7.2.2 that (M,g)isgeodesically complete iffthe exponential map, v expp(v), is defined for all p M and for all v T M.,→ ∈ ∈ p 7.3. COMPLETE RIEMANNIAN MANIFOLDS, HOPF-RINOW, CUT LOCUS 511 We now prove the following important theorem due to Hopf and Rinow (1931):

Theorem 7.3.3 (Hopf-Rinow)Let(M,g) be a con- nected Riemannian manifold. If there is a point, p M,suchthatexp is defined on the entire tangent ∈ p space, TpM,thenanypoint,q M,canbejoinedto p by a minimal geodesic. As a∈ consequence, if M is geodesically complete, then any two points of M can be joined by a minimal geodesic. Proof .ThemostbeautifulproofisMilnor’sproofin[?], Chapter 10, Theorem 10.9.

Theorem 7.3.3 implies the following result (often known as the Hopf-Rinow Theorem): 512 CHAPTER 7. GEODESICS ON RIEMANNIAN MANIFOLDS Theorem 7.3.4 Let (M,g) be a connected, Rieman- nian manifold. The following statements are equiva- lent: (1) The manifold (M,g) is geodesically complete, that is, for every p M,everygeodesicthroughp can be extended to∈ a geodesic defined on all of R. (2) For every point, p M,themapexp is defined ∈ p on the entire tangent space, TpM. (3) There is a point, p M,suchthatexp is defined ∈ p on the entire tangent space, TpM. (4) Any closed and bounded subset of the metric space, (M,d),iscompact. (5) The metric space, (M,d),iscomplete(thatis,ev- ery Cauchy sequence converges).

In view of Theorem 7.3.4, a connected Riemannian mani- fold, (M,g), is geodesically complete iffthe metric space, (M,d), is complete. 7.3. COMPLETE RIEMANNIAN MANIFOLDS, HOPF-RINOW, CUT LOCUS 513 We will refer simply to M as a complete Riemannian manifold (it is understood that M is connected).

Also, by (4), every compact, Riemannian manifold is com- plete.

If we remove any point, p,fromaRiemannianmanifold, M,thenM p is not complete since every geodesic that formerly− went{ } through p yields a geodesic that can’t be extended.

Assume (M,g) is a complete Riemannian manifold. Given any point, p M,itisinterestingtoconsiderthesub- ∈ set, p TpM,consistingofallv TpM such that the geodesicU ⊆ ∈ t exp (tv) ,→ p is a minimal geodesic up to t =1+",forsome">0. 514 CHAPTER 7. GEODESICS ON RIEMANNIAN MANIFOLDS The subset is open and star-shaped and it turns out Up that expp is a diffeomorphism from p onto its image, exp ( ), in M. U p Up The left-over part, M exp ( )(ifnonempty),isactu- − p Up ally equal to expp(∂ p)anditisanimportantsubsetof M called the cut locusU of p.

Proposition 7.3.5 Let (M,g) be a complete Rieman- nian manifold. For any geodesic, γ:[0,a] M,fromp = γ(0) to q = γ(a),thefollowing properties→ hold: (i) If there is no geodesic shorter than γ between p and q,thenγ is minimal on [0,a]. (ii) If there is another geodesic of the same length as γ between p and q,thenγ is no longer minimal on any larger interval, [0,a+ "]. (iii) If γ is minimal on any interval, I,thenγ is also minimal on any subinterval of I. 7.3. COMPLETE RIEMANNIAN MANIFOLDS, HOPF-RINOW, CUT LOCUS 515 Again, assume (M,g)isacompleteRiemannianmanifold and let p M be any point. For every v T M,let ∈ ∈ p I = s R the geodesic t exp (tv) v { ∈ ∪{∞}| ,→ p is minimal on [0,s] . }

It is easy to see that Iv is a closed interval, so Iv =[0,ρ(v)] (with ρ(v)possiblyinfinite).

It can be shown that if w = λv,thenρ(v)=λρ(w), so we can restrict our attention to unit vectors, v.

n 1 It can also be shown that the map, ρ: S − R,is n 1 → continuous, where S − is the unit sphere of center 0 in TpM,andthatρ(v)isboundedbelowbyastrictly positive number. 516 CHAPTER 7. GEODESICS ON RIEMANNIAN MANIFOLDS Definition 7.3.6 Let (M,g)beacompleteRiemannian manifold and let p M be any point. Define by ∈ Up v = v T M ρ > v Up ∈ p v " " ( ) & ' * = v T M )ρ(v)">"1 p ) { ∈ |) } and the cut locus of p by n 1 Cut(p)=exp(∂ )= exp (ρ(v)v) v S − . p Up { p | ∈ }

The set is open and star-shaped. Up The , ∂ ,of in T M is sometimes called the Up Up p tangential cut locus of p and is denoted Cut(p).

Remark: The cut locus was first introduced+ for con- vex surfaces by Poincar´e(1905) under the name ligne de partage. 7.3. COMPLETE RIEMANNIAN MANIFOLDS, HOPF-RINOW, CUT LOCUS 517 According to Do Carmo [?](Chapter13,Section2),for Riemannian manifolds, the cut locus was introduced by J.H.C. Whitehead (1935).

But it was Klingenberg (1959) who revived the interest in the cut locus and showed its usefuleness.

Proposition 7.3.7 Let (M,g) be a complete Rieman- nian manifold. For any point, p M,thesetsexpp( p) and Cut(p) are disjoint and ∈ U M =exp( ) Cut(p). p Up ∪

Observe that the injectivity radius, i(p), of M at p is equal to the distance from p to the cut locus of p: i(p)=d(p, Cut(p)) = inf d(p, q). q Cut(p) ∈

Consequently, the injectivity radius, i(M), of M is given by i(M)= inf d(p, Cut(p)). p M ∈ 518 CHAPTER 7. GEODESICS ON RIEMANNIAN MANIFOLDS If M is compact, it can be shown that i(M) > 0. It can also be shown using Jacobi fields that expp is a diffeomor- phism from onto its image, exp ( ). Up p Up n Thus, expp( p)isdiffeomorphictoanopenballinR (where n =dim(U M)) and the cut locus is closed.

Hence, the manifold, M,isobtainedbygluingtogether an open n-ball onto the cut locus of a point. In some sense the topology of M is “contained” in its cut locus.

n 1 Given any sphere, S − ,thecutlocusofanypoint,p,is its antipodal point, p . {− } In general, the cut locus is very hard to compute. In fact, according to Berger [?], even for an ellipsoid, the determination of the cut locus of an arbitrary point is still a matter of ! 7.4. THE OF VARIATIONS APPLIED TO GEODESICS 519 7.4 The Applied to Geodesics; The First Variation Formula

Given a Riemannian manifold, (M,g), the path space, Ω(p, q), was introduced in Definition 7.1.1.

It is an “infinite dimensional” manifold. By analogy with finite dimensional manifolds we define a kind of tangent space to Ω(p, q)atapointω.

In this section, it is convenient to assume that paths in Ω(p, q)areparametrizedovertheinterval[0, 1].

Definition 7.4.1 For every “point”, ω Ω(p, q), we ∈ define the “tangent space”, TωΩ(p, q), of Ω(p, q)atω, to be the space of all piecewise smooth vector fields, W , along ω,forwhichW (0) = W (1) = 0 (we may assume that our paths, ω,areparametrizedover[0, 1]). 520 CHAPTER 7. GEODESICS ON RIEMANNIAN MANIFOLDS Now, if F :Ω(p, q) R is a real-valued function on Ω(p, q), it is natural→ to ask what the induced “tangent map”, dF : T Ω(p, q) R, ω ω → should mean (here, we are identifying TF (ω)R with R).

Observe that Ω(p, q)isnotevenatopologicalspaceso the answer is far from obvious!

In the case where f: M R is a function on a mani- fold, there are various equivalent→ ways to define df ,one of which involves curves.

For every v T M,ifα:( ",") M is a curve such ∈ p − → that α(0) = p and α((0) = v,thenweknowthat d(f(α(t))) df (v)= . p dt )t=0 ) ) ) 7.4. THE CALCULUS OF VARIATIONS APPLIED TO GEODESICS 521 We may think of α as a small variation of p.Recallthat p is a critical point of f iff df (v)=0,forallv T M. p ∈ p

Rather than attempting to define dFω (which requires some conditions on F ), we will mimic what we did with functions on manifolds and define what is a critical path of a function, F :Ω(p, q) R,usingthenotionofvaria- tion. →

Now, geodesics from p to q are special paths in Ω(p, q) and they turn out to be the critical paths of the energy function, b b 2 E (ω)= ω((t) dt, a " " $a where ω Ω(p, q), and 0 a0, such that − → (1) α(0) = ω " (2) There is a subdivision, 0 = t0

The function, α,maybeconsideredasa“smoothpath” in Ω(p, q), since for every u ( ","), the map α(u)is ∈ − acurveinΩ(p," q)calledacurve in the variation (or longitudinal curve of the variation). "

The “velocity vector”, dα(0) T Ω(p, q), is defined to be du ∈ ω the vector field, W ,along" ω,givenby dα ∂α W = (0) = (0,t), t du t ∂u " Clearly, W TωΩ(p, q). In particular, W (0) = W (1)∈ = 0.

The vector field, W ,isalsocalledthevariation vector field associated with the variation α. 524 CHAPTER 7. GEODESICS ON RIEMANNIAN MANIFOLDS Besides the curves in the variation, α(u)(withu ( ",")), for every t [0, 1], we have a curve, α :( ","∈) − M, ∈ t − → called a transversal curve of the variation" ,definedby

αt(u)=α(u)(t), and Wt is equal to the velocity vector, αt((0), at the point " ω(t)=αt(0).

For " sufficiently small, the vector field, Wt,isaninfinites- imal model of the variation α.

We can show that for any W" TωΩ(p, q)thereisavari- ation, α:( ",") Ω(p, q), which∈ satisfies the conditions − → dα α(0) = ω, (0) = W. " du " " As we said earlier, given a function, F :Ω(p, q) R,we → do not attempt to define the differential, dFω,butinstead, the notion of critical path. 7.4. THE CALCULUS OF VARIATIONS APPLIED TO GEODESICS 525 Definition 7.4.3 Given a function, F :Ω(p, q) R,we say that a path, ω Ω(p, q), is a critical path for→ F iff ∈ dF (α(u)) =0, du )u=0 ) for every variation, α,of"ω (which) implies that the deriva- ) tive dF (α(u)) is defined for every variation, α,ofω). du u=0 " ) " ) ) " For example, if F takes on its minimum on a path ω0 dF (α(u)) and if the du are all defined, then ω0 is a critical path of F . "

We will apply the above to two functions defined on Ω(p, q): 526 CHAPTER 7. GEODESICS ON RIEMANNIAN MANIFOLDS (1) The energy function (also called integral): b b 2 E (ω)= ω((t) dt. a " " $a 1 (We write E = E0.) (2) The arc-length function, b b L (ω)= ω((t) dt. a " " $a

b b The quantities Ea(ω)andLa(ω)canbecomparedasfol- lows: if we apply the Cauchy-Schwarz’s inequality, 2 b b b f(t)g(t)dt f 2(t)dt (t)dt ≤ ,$a - ,$a -,$a - with f(t) 1andg(t)= ω((t) ,weget ≡ " " (Lb (ω))2 (b a)Eb, a ≤ − a where equality holds iff g is constant; that is, iffthe pa- rameter t is proportional to arc-length. 7.4. THE CALCULUS OF VARIATIONS APPLIED TO GEODESICS 527 Now, suppose that there exists a minimal geodesic, γ, from p to q.Then, E(γ)=L(γ)2 L(ω)2 E(ω), ≤ ≤ where the equality L(γ)2 = L(ω)2 holds only if ω is also aminimalgeodesic,possiblyreparametrized.

On the other hand, the equality L(ω)=E(ω)2 can hold only if the parameter is proportional to arc-length along ω.

This proves that E(γ)

Proposition 7.4.4 Let (M,g) be a complete Rieman- nian manifold. For any two points, p, q M,if d(p, q)=δ,thentheenergyfunction,E:Ω(p,∈ q) R, takes on its minimum, δ2,preciselyonthesetofmin-→ imal geodesics from p to q. 528 CHAPTER 7. GEODESICS ON RIEMANNIAN MANIFOLDS Next, we are going to show that the critical paths of the energy function are exactly the geodesics. For this, we need the first variation formula.

Let α:( ",") Ω(p, q)beavariationofω and let − → ∂α W = (0,t) " t ∂u be its associated variation vector field.

Furthermore, let dω V = = ω((t), t dt the velocity vector of ω and

∆tV = Vt+ Vt , − − the discontinuity in the velocity vector at t,whichis nonzero only for t = ti,with0

Intuitively," the first term on the right-hand side shows that varying the path ω in the direction of decreasing “kink” tends to decrease E.

The second term shows that varying the curve in the D direction of its acceleration vector, dt ω((t), also tends to reduce E.

Ageodesic,γ,(parametrizedover[0, 1]) is smooth on the D entire interval [0, 1] and its acceleration vector, dt γ((t), is identically zero along γ.Thisgivesushalfof 530 CHAPTER 7. GEODESICS ON RIEMANNIAN MANIFOLDS Theorem 7.4.6 Let (M,g) be a Riemanian manifold. For any two points, p, q M,apath,ω Ω(p, q) (parametrized over [0, 1]),∈ is critical for the∈ energy function, E,iffω is a geodesic.

Remark: If ω Ω(p, q)isparametrizedbyarc-length, it is easy to prove∈ that dL(α(u)) 1 dE(α(u)) = . du 2 du )u=0 )u=0 ) ) " ) " ) As a consequence, a) path, ω Ω(p, q)iscriticalforthe) arc-length function, L,iffitcanbereparametrizedsothat∈ it is a geodesic

In order to go deeper into the study of geodesics we need Jacobi fields and the “second variation formula”, both involving a term.

Therefore, we now proceed with a more thorough study of curvature on Riemannian manifolds.